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The four exponentials conjecture asserts that if $x_1,x_2$ are complex numbers linearly independent over $\mathbb{Q}$, and $y_1,y_2$ are likewise linearly independent over $\mathbb{Q}$, then at least one of the four numbers $e^{x_i{y}_j}$ is a transcendental number.

Both independence hypotheses are essential to the statement. It is a problem in transcendental number theory, asking how a two-by-two array of products constrains exponentiation. The analogous statement with two independent $x_i$ and three independent $y_j$ is the six exponentials theorem; that stronger supply of inputs is known to force at least one transcendental output. The four-number assertion is a conjecture, not a consequence of checking individual exponentials.

Transcendental number theory studies numbers that satisfy no nonzero polynomial equation with integer coefficients, and stronger independence properties among numbers and special-function values. The distinction is algebraic: $\sqrt{2}$ is irrational but algebraic because it satisfies $x^2-2=0$.

A useful example is the Liouville constant $L={\sum }_{n\ge 1}10^{-n!}$. Truncating after $N$ terms gives a rational $p_N/q_N$ with $q_N=10^{N!}$ andFor any fixed algebraic degree, these approximations eventually contradict the rational-approximation lower bound in Liouville approximation theorem. Thus exceptionally good approximation can certify transcendence.

Other results, such as the six exponentials theorem, constrain collections of exponential values using linear independence of their inputs. Proving each number irrational is weaker than proving it transcendental, and proving separate transcendence is weaker than proving algebraic independence of a collection.